We are given that $\angle MYT \cong \angle NYT$ and $\angle MTY \cong \angle NTY$. We need to prove that $\triangle RYM \cong \triangle RYN$.

1. Problem Description

We are given that $\angle MYT \cong \angle NYT$ and $\angle MTY \cong \angle NTY$. We need to prove that $\triangle RYM \cong \triangle RYN$.

2. Solution Steps

Statement | Reason

------- | --------

1. $\angle MYT \cong \angle NYT$ |

1. Given

2. $\angle MTY \cong \angle NTY$ |

2. Given

3. $YT \cong YT$ |

3. Reflexive Property of Congruence

4. $\triangle MYT \cong \triangle NYT$ |

4. Angle-Side-Angle (ASA) Congruence Theorem (using statements 1, 2, and 3)

5. $MY \cong NY$ |

5. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

6. $\angle RYM \cong \angle RYN$ |

6. $\angle MYT$ and $\angle RYM$ are supplementary angles. $\angle NYT$ and $\angle RYN$ are supplementary angles. If two angles are congruent, then their supplements are congruent.

7. $RY \cong RY$ |

7. Reflexive Property of Congruence

8. $\triangle RYM \cong \triangle RYN$ |

8. Side-Angle-Side (SAS) Congruence Theorem (using statements 5, 6, and 7)

3. Final Answer

$\triangle RYM \cong \triangle RYN$